Fall 2020
Math 4330: General Topology



Course format

This class will be offered entirely online. We will meet live during class time over Zoom, where (on top of traditional lecturing activities) we will engage in group work and you will be able to ask questions. To the extent possible, instructor will upload lecture videos and class notes after every class. Office hours will take place online.

If you are taking this class, but have any concerns about accessibility--including to fast-enough internet, to technology (such as a laptop), and to a quiet location where you can attend lecture--let Hiro know as soon as possible.

Tu/Th: 2 PM - 3:20 PM.

Course description

This class will be an introduction to the basic language necessary to study topology. Some buzz-words you can Google include: topological spaces, continuity, compactness, metric spaces, connectedness, and Hausdorffness.

Beyond a fluency with the above topics, another goal of this class is for you to become familiar with mathematical thinking---questioning and understanding why definitions exist, identifying when you or another communicator is being precise or imprecise (and for what purpose), developing tastes that are rooted in practice and informed experience, exploring the mathematical landscape on your own.

Prerequisite: MATH 3330 or MATH 3380 with a grade of C or higher.


Textbook and resources.

You do not need to buy a textbook for this class. The following are freely available resources:

  1. Here is a link to the Fall 2019 version of this course. There you will find past course notes, past homework assignments, et cetera.
  2. Allen Hatcher's notes on point-set topology.
  3. Stefan Waner's notes on Elementary Topology.
  4. Sidney A. Morris's Topology without Tears.

Resources for proof (all freely available):

  1. Book of Proof by Richard Hammack. You will need to be comfortable with all the material in this book.
  2. Introduction to Proof in Analysis by Steve Halperin. You will need to be comfortable with Chapters 1-3 of this book.
  3. Introduction to mathematical arguments by Michael Hutchings. You will need to be comfortable with all this material.



The syllabus

This is the course syllabus as of August 23rd.

The survey

Here is the survey. Please fill out by Thursday, August 27th, at 11:59 PM. It should take no more than 45 minutes. Names will be removed from the survey responses, but all other results of the survey will be shared with the class.

Important dates

Midterm Exam (Exam I): Tue Oct 6 (online, in-class)
Final Exam: Tue Dec 8, 2 - 4:30 PM (online)

Collaboration policy

I strongly encourage all of you to collaborate. Please do so. If you do, you must indicate clearly on every assignment that you have collaborated, and indicate with whom. However, write solutions on your own. It is fine to think through problems and find solutions with each other, but when it comes to the act of writing your homework, you must do so without assistance from another. This is because the act of solving something and writing a mathematical proof are two different skills, and I want you to also hone the latter. As an extreme anti-example, copying and pasting solutions/proofs will not be tolerated. To reiterate, you may not write solutions together.

Recordings of Zoom Lectures

These may be found here. To protect student privacy, these are only accessible via Texas State University NetID.




Notes

  1. Tue, Aug 25. Sets and power sets.
  2. Thu, Aug 27. Important sets in topology.
  3. Tue, Sep 1. Posets.
  4. Thu, Sep 3. Every poset is inside a power set. Unions and intersections.
  5. Tue, Sep 8. Open subsets of Euclidean space.
  6. Thu, Sep 10. Closed sets and practice.
  7. Tue, Sep 15. Topologies and continuity.
  8. Thu, Sep 17. Subspaces.
  9. Tue, Sep 22. Compactness, I.
  10. Thu, Sep 24. Compactness, II. Heine-Borel Theorem.
  11. Tue, Sep 29. Compactness, III. Extreme Value Theorem.
    Thu, Oct 1. Questions and Answer Session before Exam.
    Tue, Oct 6. Midterm Exam
  12. Thu, Oct 8. Equivalence relations and quotient sets.
  13. Tue, Oct 13. Quotient spaces.
  14. Thu, Oct 15. Universal property of quotient spaces. (Using same notes and exercise as previous lecture.)
  15. Tue, Oct 20. Product spaces and their universal property.
  16. Thu, Oct 22. Same topic and exercises as last time (using same notes and exercise as previous lecture).
  17. Tue, Oct 27. The Hausdorff property.
  18. Thu, Oct 29. Metric spaces.
  19. Tue, Nov 3. Isometries.
  20. Thu, Nov 5. Path-connectedness.
  21. Tue, Nov 10. Invariance of domain and pi_0.
  22. Thu, Nov 12. Connectedness, stereographic projection, one-point compactification.
  23. Tue, Nov 17. Interior/Closure/Neighborhoods/Density.
  24. Thu, Nov 19. Exercise Day.
  25. Tue, Nov 24. Brouwer Fixed Point Theorem and other fun topics.
    Thu, Nov 26. Thanksgiving, no class.
  26. Tue, Dec 1. Review Q+A for Final.
  27. Thu, Dec 3. Last day of class. What comes next?


Homeworks

Make sure to fill out the survey by Thursday, August 27th, at 11:59 PM.
All assignments are due on Canvas, and must be uploaded in PDF format. All true/false questions are also to be answered at the corresponding quiz on Canvas.
All homework assignments will be shared with your classmates, so you may remove your names from your scanned/typed/written assignments. (When you upload your homework, I will know which assignments belong to whom, thanks to Canvas.)
  1. Writing 1. Due Friday, August 28, 11:59 PM.
    Homework 1. Due Tuesday, September 1, 11:59 PM.
    Extra Credit 1. Due Thursday, September 3, 11:59 PM.
  2. Writing 2. Due Friday, September 4, 11:59 PM.
    Homework 2. Due Tuesday, September 8, 11:59 PM.
    Extra Credit 2. Due Thursday, September 10, 11:59 PM.
  3. Writing 3. Due Friday, September 11, 11:59 PM.
    Homework 3. Due Tuesday, September 15, 11:59 PM.
    Extra Credit 3. Due Thursday, September 17, 11:59 PM.
  4. Writing 4. Due Friday, September 18, 11:59 PM.
    Homework 4. Due Tuesday, September 22, 11:59 PM.
    Extra Credit 4. Due Thursday, September 24, 11:59 PM.
  5. Writing 5. Due Friday, September 25, 11:59 PM.
    Homework 5. Due Tuesday, September 29, 11:59 PM.
    Extra Credit 5. Due Thursday, October 1, 11:59 PM.
  6. Writing 6. Due Friday, October 2, 11:59 PM.
    Homework 6. Due Tuesday, October 6, 11:59 PM. (The same day as the exam!)
    Extra Credit 6. Due Thursday, October 8, 11:59 PM.
  7. Writing 7. Due Friday, October 9, 11:59 PM.
    Homework 7. Due Tuesday, October 13, 11:59 PM.
    Extra Credit 7. Due Thursday, October 15, 11:59 PM.
  8. Writing 8. Due Friday, October 16, 11:59 PM.
    Homework 8. Due Tuesday, October 20, 11:59 PM.
    Extra Credit 8. Due Thursday, October 22, 11:59 PM.
  9. Writing 9. Due Friday, October 23, 11:59 PM.
    Homework 9. Due Tuesday, October 27, 11:59 PM.
    Extra Credit 9. Due Thursday, October 29, 11:59 PM.
  10. Writing 10. Due Friday, October 30, 11:59 PM.
    Homework 10. Due Wednesday, November 4 (Due date extended by one day), 11:59 PM.
    Extra Credit 10. Due Thursday, November 5, 11:59 PM.
  11. Writing 11. Due Friday, November 6, 11:59 PM.
    Homework 11. Due Tuesday, November 10, 11:59 PM.
    Extra Credit 11. Due Thursday, November 12, 11:59 PM.
  12. Writing 12. Due Friday, November 13, 11:59 PM.
    Homework 12. Due Tuesday, November 17, 11:59 PM.
    Extra Credit 12. Due Thursday, November 19, 11:59 PM.
  13. Writing 13. Due Friday, November 20, 11:59 PM.
    Homework 13. Due Tuesday, November 24, 11:59 PM.
    Extra Credit 13 (last one, many parts!). Due Wednesday, December 2, 11:59 PM.
  14. Writing 14. (You have extra time due to Thanksgiving.) Due Monday, November 30, 11:59 PM.
    Homework 14. Due Tuesday, December 1, 11:59 PM.